MODULE grid_interpolation

!      ALGORITHM 760, COLLECTED ALGORITHMS FROM ACM.
!      THIS WORK PUBLISHED IN TRANSACTIONS ON MATHEMATICAL SOFTWARE,
!      VOL. 22, NO. 3, September, 1996, P.  357--361.

IMPLICIT NONE
PRIVATE
PUBLIC  :: rgbi3p, rgsf3p


CONTAINS


SUBROUTINE rgbi3p(md, nxd, nyd, xd, yd, zd, nip, xi, yi, zi, ier)
 
! Code converted using TO_F90 by Alan Miller
! Date: 2003-06-11  Time: 10:11:03

! Rectangular-grid bivariate interpolation
! (a master subroutine of the RGBI3P/RGSF3P subroutine package)

! Hiroshi Akima
! U.S. Department of Commerce, NTIA/ITS
! Version of 1995/08

! This subroutine performs interpolation of a bivariate function, z(x,y), on a
! rectangular grid in the x-y plane.  It is based on the revised Akima method.

! In this subroutine, the interpolating function is a piecewise function
! composed of a set of bicubic (bivariate third-degree) polynomials, each
! applicable to a rectangle of the input grid in the x-y plane.
! Each polynomial is determined locally.

! This subroutine has the accuracy of a bicubic polynomial, i.e., it
! interpolates accurately when all data points lie on a surface of a
! bicubic polynomial.

! The grid lines can be unevenly spaced.

!(md, nxd, nyd, xd, yd, zd, nip, xi, yi, zi, ier)
! The input arguments are
!   MD  = mode of computation
!       = 1 for new XD, YD, or ZD data (default)
!       = 2 for old XD, YD, and ZD data,
!   NXD = number of the input-grid data points in the x coordinate
!         (must be 2 or greater),
!   NYD = number of the input-grid data points in the y coordinate
!         (must be 2 or greater),
!   XD  = array of dimension NXD containing the x coordinates of the
!         input-grid data points (must be in a monotonic increasing order),
!   YD  = array of dimension NYD containing the y coordinates of the
!         input-grid data points (must be in a monotonic increasing order),
!   ZD  = two-dimensional array of dimension NXD*NYD
!         containing the z(x,y) values at the input-grid data points,
!   NIP = number of the output points at which interpolation
!         of the z value is desired (must be 1 or greater),
!   XI  = array of dimension NIP containing the x coordinates
!         of the output points,
!   YI  = array of dimension NIP containing the y coordinates
!         of the output points.

! The output arguments are
!   ZI  = array of dimension NIP where the interpolated z
!         values at the output points are to be stored,
!   IER = error flag
!       = 0 for no errors
!       = 1 for NXD = 1 or less
!       = 2 for NYD = 1 or less
!       = 3 for identical XD values or XD values out of sequence
!       = 4 for identical YD values or YD values out of sequence
!       = 5 for NIP = 0 or less.

! N.B. The workspace has been removed from the argument list.
!   WK  = three dimensional array of dimension 3*NXD*NYD used internally
!         as a work area.

! The very fisrt call to this subroutine and the call with a new XD, YD, and
! ZD array must be made with MD=1.  The call with MD=2 must be preceded by
! another call with the same XD, YD, and ZD arrays.  Between the call with
! MD=2 and its preceding call, the WK array must not be disturbed.

! The constant in the PARAMETER statement below is
!   NIPIMX = maximum number of output points to be processed at a time.
! The constant value has been selected empirically.

! This subroutine calls the RGPD3P, RGLCTN, and RGPLNL subroutines.


! Specification statements
!     .. Parameters ..

INTEGER, INTENT(IN)   :: md
INTEGER, INTENT(IN)   :: nxd
INTEGER, INTENT(IN)   :: nyd
DOUBLE PRECISION, INTENT(IN)      :: xd(nxd)
DOUBLE PRECISION, INTENT(IN)      :: yd(nyd)
DOUBLE PRECISION, INTENT(IN OUT)  :: zd(nxd,nyd)
INTEGER, INTENT(IN)   :: nip
DOUBLE PRECISION, INTENT(IN OUT)  :: xi(nip)
DOUBLE PRECISION, INTENT(IN OUT)  :: yi(nip)
DOUBLE PRECISION, INTENT(IN OUT)  :: zi(nip)
INTEGER, INTENT(OUT)  :: ier

!     ..
!     .. Local Scalars ..
INTEGER, PARAMETER  :: nipimx=51

INTEGER  :: iip, ix, iy, nipi
!     ..
!     .. Local Arrays ..
INTEGER  :: inxi(nipimx), inyi(nipimx)

! Allocate workspace
DOUBLE PRECISION  :: wk(3,nxd,nyd)
!     ..
!     .. External Subroutines ..
! EXTERNAL         rglctn, rgpd3p, rgplnl
!     ..
!     .. Intrinsic Functions ..
! INTRINSIC        MIN
!     ..

! Preliminary processing
! Error check
IF (nxd <= 1) GO TO 40
IF (nyd <= 1) GO TO 50
DO  ix = 2,nxd
  IF (xd(ix) <= xd(ix-1)) GO TO 60
END DO
DO  iy = 2,nyd
  IF (yd(iy) <= yd(iy-1)) GO TO 70
END DO
IF (nip <= 0) GO TO 80
ier = 0

! Calculation
! Estimates partial derivatives at all input-grid data points (for MD=1).
IF (md /= 2) THEN
  CALL rgpd3p(nxd, nyd, xd, yd, zd, wk)
END IF

! DO-loop with respect to the output point
! Processes NIPIMX output points, at most, at a time.
DO  iip = 1,nip,nipimx
  nipi = MIN(nip- (iip-1),nipimx)
! Locates the output points.
  CALL rglctn(nxd, nyd, xd, yd, nipi, xi(iip), yi(iip), inxi, inyi)

! Calculates the z values at the output points.
  CALL rgplnl(nxd, nyd, xd, yd, zd, wk, nipi, xi(iip), yi(iip), inxi, inyi, &
              zi(iip))
END DO
RETURN

! Error exit
40 WRITE (*,FMT=9000)
ier = 1
GO TO 90
50 WRITE (*,FMT=9010)
ier = 2
GO TO 90
60 WRITE (*,FMT=9020) ix,xd(ix)
ier = 3
GO TO 90
70 WRITE (*,FMT=9030) iy,yd(iy)
ier = 4
GO TO 90
80 WRITE (*,FMT=9040)
ier = 5
90 WRITE (*,FMT=9050) nxd,nyd,nip
RETURN

! Format statements for error messages
9000 FORMAT (/' *** RGBI3P Error 1: NXD = 1 or less')
9010 FORMAT (/' *** RGBI3P Error 2: NYD = 1 or less')
9020 FORMAT (/' *** RGBI3P Error 3: Identical XD values or',  &
    ' XD values out of sequence'/ '    IX =', i6, ',  XD(IX) =', e11.3)
9030 FORMAT (/' *** RGBI3P Error 4: Identical YD values or',  &
    ' YD values out of sequence',/,'    IY =',i6,',  YD(IY) =', e11.3)
9040 FORMAT (/' *** RGBI3P Error 5: NIP = 0 or less')
9050 FORMAT ('    NXD =', i5,',  NYD =', i5,',  NIP =', i5/)
END SUBROUTINE rgbi3p



SUBROUTINE rgsf3p(md, nxd, nyd, xd, yd, zd, nxi, xi, nyi, yi, zi, ier)

! Rectangular-grid surface fitting
! (a master subroutine of the RGBI3P/RGSF3P subroutine package)

! Hiroshi Akima
! U.S. Department of Commerce, NTIA/ITS
! Version of 1995/08

! This subroutine performs surface fitting by interpolating values of a
! bivariate function, z(x,y), on a rectangular grid in the x-y plane.
! It is based on the revised Akima method.

! In this subroutine, the interpolating function is a piecewise function
! composed of a set of bicubic (bivariate third-degree) polynomials, each
! applicable to a rectangle of the input grid in the x-y plane.
! Each polynomial is determined locally.

! This subroutine has the accuracy of a bicubic polynomial, i.e., it fits the
! surface accurately when all data points lie on a surface of a bicubic
! polynomial.

! The grid lines of the input and output data can be unevenly spaced.

! The input arguments are
!   MD  = mode of computation
!       = 1 for new XD, YD, or ZD data (default)
!       = 2 for old XD, YD, and ZD data,
!   NXD = number of the input-grid data points in the x
!         coordinate (must be 2 or greater),
!   NYD = number of the input-grid data points in the y
!         coordinate (must be 2 or greater),
!   XD  = array of dimension NXD containing the x coordinates
!         of the input-grid data points (must be in a
!         monotonic increasing order),
!   YD  = array of dimension NYD containing the y coordinates
!         of the input-grid data points (must be in a
!         monotonic increasing order),
!   ZD  = two-dimensional array of dimension NXD*NYD
!         containing the z(x,y) values at the input-grid data points,
!   NXI = number of output grid points in the x coordinate
!         (must be 1 or greater),
!   XI  = array of dimension NXI containing the x coordinates
!         of the output grid points,
!   NYI = number of output grid points in the y coordinate
!         (must be 1 or greater),
!   YI  = array of dimension NYI containing the y coordinates
!         of the output grid points.

! The output arguments are
!   ZI  = two-dimensional array of dimension NXI*NYI, where the interpolated
!         z values at the output grid points are to be stored,
!   IER = error flag
!       = 0 for no error
!       = 1 for NXD = 1 or less
!       = 2 for NYD = 1 or less
!       = 3 for identical XD values or XD values out of sequence
!       = 4 for identical YD values or YD values out of sequence
!       = 5 for NXI = 0 or less
!       = 6 for NYI = 0 or less.

! N.B. The workspace has been removed from the argument list.
!   WK  = three-dimensional array of dimension 3*NXD*NYD used internally
!         as a work area.

! The very first call to this subroutine and the call with a new XD, YD, or
! ZD array must be made with MD=1.  The call with MD=2 must be preceded by
! another call with the same XD, YD, and ZD arrays.  Between the call with
! MD=2 and its preceding call, the WK array must not be disturbed.

! The constant in the PARAMETER statement below is
!   NIPIMX = maximum number of output points to be processed at a time.
! The constant value has been selected empirically.

! This subroutine calls the RGPD3P, RGLCTN, and RGPLNL subroutines.


! Specification statements
!     .. Parameters ..

INTEGER, INTENT(IN)   :: md
INTEGER, INTENT(IN)   :: nxd
INTEGER, INTENT(IN)   :: nyd
DOUBLE PRECISION, INTENT(IN)      :: xd(nxd)
DOUBLE PRECISION, INTENT(IN)      :: yd(nyd)
DOUBLE PRECISION, INTENT(IN OUT)  :: zd(nxd,nyd)
INTEGER, INTENT(IN)   :: nxi
DOUBLE PRECISION, INTENT(IN OUT)  :: xi(nxi)
INTEGER, INTENT(IN)   :: nyi
DOUBLE PRECISION, INTENT(IN)      :: yi(nyi)
DOUBLE PRECISION, INTENT(IN OUT)  :: zi(nxi,nyi)
INTEGER, INTENT(OUT)  :: ier

!     ..
!     .. Local Scalars ..
INTEGER, PARAMETER  :: nipimx=51

INTEGER  :: ix, ixi, iy, iyi, nipi
!     ..
!     .. Local Arrays ..
DOUBLE PRECISION     :: yii(nipimx)
INTEGER  :: inxi(nipimx), inyi(nipimx)

! Allocate workspace
DOUBLE PRECISION  :: wk(3,nxd,nyd)
!     ..
!     .. External Subroutines ..
! EXTERNAL         rglctn,rgpd3p,rgplnl
!     ..
!     .. Intrinsic Functions ..
! INTRINSIC        MIN
!     ..

! Preliminary processing
! Error check
IF (nxd <= 1) GO TO 60
IF (nyd <= 1) GO TO 70
DO  ix = 2,nxd
  IF (xd(ix) <= xd(ix-1)) GO TO 80
END DO
DO  iy = 2,nyd
  IF (yd(iy) <= yd(iy-1)) GO TO 90
END DO
IF (nxi <= 0) GO TO 100
IF (nyi <= 0) GO TO 110
ier = 0

! Calculation
! Estimates partial derivatives at all input-grid data points
! (for MD=1).
IF (md /= 2) THEN
  CALL rgpd3p(nxd, nyd, xd, yd, zd, wk)
END IF

! Outermost DO-loop with respect to the y coordinate of the output grid points
DO  iyi = 1,nyi
  DO  ixi = 1,nipimx
    yii(ixi) = yi(iyi)
  END DO

! Second DO-loop with respect to the x coordinate of the output grid points
! Processes NIPIMX output-grid points, at most, at a time.
  DO  ixi = 1,nxi,nipimx
    nipi = MIN(nxi- (ixi-1), nipimx)
! Locates the output-grid points.
    CALL rglctn(nxd, nyd, xd, yd, nipi, xi(ixi), yii, inxi, inyi)

! Calculates the z values at the output-grid points.
    CALL rgplnl(nxd, nyd, xd, yd, zd, wk, nipi, xi(ixi), yii, inxi, inyi,  &
                zi(ixi,iyi))
  END DO
END DO
RETURN

! Error exit
60 WRITE (*,FMT=9000)
ier = 1
GO TO 120
70 WRITE (*,FMT=9010)
ier = 2
GO TO 120
80 WRITE (*,FMT=9020) ix,xd(ix)
ier = 3
GO TO 120
90 WRITE (*,FMT=9030) iy,yd(iy)
ier = 4
GO TO 120
100 WRITE (*,FMT=9040)
ier = 5
GO TO 120
110 WRITE (*,FMT=9050)
ier = 6
120 WRITE (*,FMT=9060) nxd,nyd,nxi,nyi
RETURN

! Format statements for error messages
9000 FORMAT (/' *** RGSF3P Error 1: NXD = 1 or less')
9010 FORMAT (/' *** RGSF3P Error 2: NYD = 1 or less')
9020 FORMAT (/' *** RGSF3P Error 3: Identical XD values or',  &
    ' XD values out of sequence',/,'    IX =',i6,',  XD(IX) =', e11.3)
9030 FORMAT (/' *** RGSF3P Error 4: Identical YD values or',  &
    ' YD values out of sequence',/,'    IY =',i6,',  YD(IY) =', e11.3)
9040 FORMAT (/' *** RGSF3P Error 5: NXI = 0 or less')
9050 FORMAT (/' *** RGSF3P Error 6: NYI = 0 or less')
9060 FORMAT ('    NXD =', i5, ',  NYD =', i5, ',  NXI =', i5,',  NYI =', i5 /)
END SUBROUTINE rgsf3p



!     ..
! Statement Function definitions
! z2f(xx1,xx2,zz0,zz1) = (zz1-zz0)*xx2/xx1 + zz0
! z3f(xx1,xx2,xx3,zz0,zz1,zz2) = ((zz2-zz0)* (xx3-xx1)/xx2 -  &
!    (zz1-zz0)* (xx3-xx2)/xx1)* (xx3/ (xx2-xx1)) + zz0

FUNCTION z2f(xx1, xx2, zz0, zz1) RESULT(fn_val)

DOUBLE PRECISION, INTENT(IN)  :: xx1, xx2, zz0, zz1
DOUBLE PRECISION              :: fn_val

fn_val = (zz1-zz0)*xx2/xx1 + zz0
RETURN
END FUNCTION z2f



FUNCTION z3f(xx1, xx2, xx3, zz0, zz1, zz2) RESULT(fn_val)

DOUBLE PRECISION, INTENT(IN)  :: xx1, xx2, xx3, zz0, zz1, zz2
DOUBLE PRECISION              :: fn_val

fn_val = ((zz2-zz0)*(xx3-xx1)/xx2 - (zz1-zz0)*(xx3-xx2)/xx1) *  &
          (xx3/(xx2-xx1)) + zz0
RETURN
END FUNCTION z3f



SUBROUTINE rgpd3p(nxd, nyd, xd, yd, zd, pdd)

! Partial derivatives of a bivariate function on a rectangular grid
! (a supporting subroutine of the RGBI3P/RGSF3P subroutine package)

! Hiroshi Akima
! U.S. Department of Commerce, NTIA/ITS
! Version of 1995/08

! This subroutine estimates three partial derivatives, zx, zy, and
! zxy, of a bivariate function, z(x,y), on a rectangular grid in
! the x-y plane.  It is based on the revised Akima method that has
! the accuracy of a bicubic polynomial.

! The input arguments are
!   NXD = number of the input-grid data points in the x
!         coordinate (must be 2 or greater),
!   NYD = number of the input-grid data points in the y
!         coordinate (must be 2 or greater),
!   XD  = array of dimension NXD containing the x coordinates of the
!         input-grid data points (must be in a monotonic increasing order),
!   YD  = array of dimension NYD containing the y coordinates of the
!         input-grid data points (must be in a monotonic increasing order),
!   ZD  = two-dimensional array of dimension NXD*NYD
!         containing the z(x,y) values at the input-grid data points.

! The output argument is
!   PDD = three-dimensional array of dimension 3*NXD*NYD,
!         where the estimated zx, zy, and zxy values at the
!         input-grid data points are to be stored.


! Specification statements
!     .. Scalar Arguments ..

INTEGER, INTENT(IN)  :: nxd
INTEGER, INTENT(IN)  :: nyd
DOUBLE PRECISION, INTENT(IN)     :: xd(nxd)
DOUBLE PRECISION, INTENT(IN)     :: yd(nyd)
DOUBLE PRECISION, INTENT(IN)     :: zd(nxd,nyd)
DOUBLE PRECISION, INTENT(OUT)    :: pdd(3,nxd,nyd)

!     ..
!     .. Local Scalars ..
DOUBLE PRECISION :: b00, b00x, b00y, b01, b10, b11, cx1, cx2, cx3, cy1, cy2,  &
        cy3, disf, dnm, dz00, dz01, dz02, dz03, dz10, dz11, dz12,  &
        dz13, dz20, dz21, dz22, dz23, dz30, dz31, dz32, dz33,  &
        dzx10, dzx20, dzx30, dzxy11, dzxy12, dzxy13, dzxy21,  &
        dzxy22, dzxy23, dzxy31, dzxy32, dzxy33, dzy01, dzy02,  &
        dzy03, epsln, pezx, pezxy, pezy, smpef, smpei, smwtf,  &
        smwti, sx, sxx, sxxy, sxxyy, sxy, sxyy, sxyz, sxz, sy, syy,  &
        syz, sz, volf, wt, x0, x1, x2, x3, y0, y1, y2,  &
        y3, z00, z01, z02, z03, z10, z11, z12, z13, z20, z21, z22,  &
        z23, z30, z31, z32, z33, zxdi, zxydi, zydi
INTEGER :: ipex, ipey, ix0, ix1, ix2, ix3, iy0, iy1, iy2, iy3, nx0, ny0
!     ..
!     .. Local Arrays ..
DOUBLE PRECISION    :: b00xa(4), b00ya(4), b01a(4), b10a(4), cxa(3,4), cya(3,4),   &
           sxa(4), sxxa(4), sya(4), syya(4), xa(3,4), ya(3,4),   &
           z0ia(3,4), zi0a(3,4)
! INTEGER :: idlt(3,4)
!     ..
!     .. Intrinsic Functions ..
! INTRINSIC        MAX
!     ..
!     .. Statement Functions ..
! DOUBLE PRECISION :: z2f,z3f
!     ..
! Data statements
! DATA ((idlt(jxy,jpexy),jpexy=1,4),jxy=1,3)/-3,-2,-1,1,-2,-1,1,2,-1,1,2,3/
INTEGER, SAVE  :: idlt(3,4) = RESHAPE(   &
                  (/ -3,-2,-1, 1,-2,-1, 1,2,-1, 1,2,3 /), (/ 3, 4 /) )
!     ..

! Calculation
! Initial setting of some local variables
nx0 = MAX(4,nxd)
ny0 = MAX(4,nyd)

! Double DO-loop with respect to the input grid points
DO  iy0 = 1,nyd
  DO  ix0 = 1,nxd
    x0 = xd(ix0)
    y0 = yd(iy0)
    z00 = zd(ix0,iy0)

! Part 1.  Estimation of ZXDI
! Initial setting
    smpef = 0.0
    smwtf = 0.0
    smpei = 0.0
    smwti = 0.0
! DO-loop with respect to the primary estimate
    DO  ipex = 1,4
! Selects necessary grid points in the x direction.
      ix1 = ix0 + idlt(1,ipex)
      ix2 = ix0 + idlt(2,ipex)
      ix3 = ix0 + idlt(3,ipex)
      IF ((ix1 < 1) .OR. (ix2 < 1) .OR. (ix3 < 1) .OR.  &
          (ix1 > nx0) .OR. (ix2 > nx0) .OR. (ix3 > nx0)) CYCLE
! Selects and/or supplements the x and z values.
      x1 = xd(ix1) - x0
      z10 = zd(ix1,iy0)
      IF (nxd >= 4) THEN
        x2 = xd(ix2) - x0
        x3 = xd(ix3) - x0
        z20 = zd(ix2,iy0)
        z30 = zd(ix3,iy0)
      ELSE IF (nxd == 3) THEN
        x2 = xd(ix2) - x0
        z20 = zd(ix2,iy0)
        x3 = 2*xd(3) - xd(2) - x0
        z30 = z3f(x1,x2,x3,z00,z10,z20)
      ELSE IF (nxd == 2) THEN
        x2 = 2*xd(2) - xd(1) - x0
        z20 = z2f(x1,x2,z00,z10)
        x3 = 2*xd(1) - xd(2) - x0
        z30 = z2f(x1,x3,z00,z10)
      END IF
      dzx10 = (z10-z00)/x1
      dzx20 = (z20-z00)/x2
      dzx30 = (z30-z00)/x3
! Calculates the primary estimate of partial derivative zx as
! the coefficient of the bicubic polynomial.
      cx1 = x2*x3/ ((x1-x2)* (x1-x3))
      cx2 = x3*x1/ ((x2-x3)* (x2-x1))
      cx3 = x1*x2/ ((x3-x1)* (x3-x2))
      pezx = cx1*dzx10 + cx2*dzx20 + cx3*dzx30
! Calculates the volatility factor and distance factor in the x
! direction for the primary estimate of zx.
      sx = x1 + x2 + x3
      sz = z00 + z10 + z20 + z30
      sxx = x1*x1 + x2*x2 + x3*x3
      sxz = x1*z10 + x2*z20 + x3*z30
      dnm = 4.0*sxx - sx*sx
      b00 = (sxx*sz-sx*sxz)/dnm
      b10 = (4.0*sxz-sx*sz)/dnm
      dz00 = z00 - b00
      dz10 = z10 - (b00+b10*x1)
      dz20 = z20 - (b00+b10*x2)
      dz30 = z30 - (b00+b10*x3)
      volf = dz00**2 + dz10**2 + dz20**2 + dz30**2
      disf = sxx

! Calculates the EPSLN value, which is used to decide whether or
! not the volatility factor is essentially zero.
      epsln = (z00**2+z10**2+z20**2+z30**2)*1.0E-12
! Accumulates the weighted primary estimates of zx and their weights.
      IF (volf > epsln) THEN
! - For a finite weight.
        wt = 1.0/ (volf*disf)
        smpef = smpef + wt*pezx
        smwtf = smwtf + wt
      ELSE
! - For an infinite weight.
        smpei = smpei + pezx
        smwti = smwti + 1.0
      END IF

! Saves the necessary values for estimating zxy
      xa(1,ipex) = x1
      xa(2,ipex) = x2
      xa(3,ipex) = x3
      zi0a(1,ipex) = z10
      zi0a(2,ipex) = z20
      zi0a(3,ipex) = z30
      cxa(1,ipex) = cx1
      cxa(2,ipex) = cx2
      cxa(3,ipex) = cx3
      sxa(ipex) = sx
      sxxa(ipex) = sxx
      b00xa(ipex) = b00
      b10a(ipex) = b10
    END DO

! Calculates the final estimate of zx.
    IF (smwti < 0.5) THEN
! - When no infinite weights exist.
      zxdi = smpef/smwtf
    ELSE
! - When infinite weights exist.
      zxdi = smpei/smwti
    END IF
! End of Part 1.

! Part 2.  Estimation of ZYDI
! Initial setting
    smpef = 0.0
    smwtf = 0.0
    smpei = 0.0
    smwti = 0.0
! DO-loop with respect to the primary estimate
    DO  ipey = 1,4
! Selects necessary grid points in the y direction.
      iy1 = iy0 + idlt(1,ipey)
      iy2 = iy0 + idlt(2,ipey)
      iy3 = iy0 + idlt(3,ipey)
      IF ((iy1 < 1) .OR. (iy2 < 1) .OR. (iy3 < 1) .OR.  &
          (iy1 > ny0) .OR. (iy2 > ny0) .OR. (iy3 > ny0)) CYCLE
! Selects and/or supplements the y and z values.
      y1 = yd(iy1) - y0
      z01 = zd(ix0,iy1)
      IF (nyd >= 4) THEN
        y2 = yd(iy2) - y0
        y3 = yd(iy3) - y0
        z02 = zd(ix0,iy2)
        z03 = zd(ix0,iy3)
      ELSE IF (nyd == 3) THEN
        y2 = yd(iy2) - y0
        z02 = zd(ix0,iy2)
        y3 = 2*yd(3) - yd(2) - y0
        z03 = z3f(y1,y2,y3,z00,z01,z02)
      ELSE IF (nyd == 2) THEN
        y2 = 2*yd(2) - yd(1) - y0
        z02 = z2f(y1,y2,z00,z01)
        y3 = 2*yd(1) - yd(2) - y0
        z03 = z2f(y1,y3,z00,z01)
      END IF
      dzy01 = (z01-z00)/y1
      dzy02 = (z02-z00)/y2
      dzy03 = (z03-z00)/y3
! Calculates the primary estimate of partial derivative zy as
! the coefficient of the bicubic polynomial.
      cy1 = y2*y3/ ((y1-y2)* (y1-y3))
      cy2 = y3*y1/ ((y2-y3)* (y2-y1))
      cy3 = y1*y2/ ((y3-y1)* (y3-y2))
      pezy = cy1*dzy01 + cy2*dzy02 + cy3*dzy03
! Calculates the volatility factor and distance factor in the y
! direction for the primary estimate of zy.
      sy = y1 + y2 + y3
      sz = z00 + z01 + z02 + z03
      syy = y1*y1 + y2*y2 + y3*y3
      syz = y1*z01 + y2*z02 + y3*z03
      dnm = 4.0*syy - sy*sy
      b00 = (syy*sz-sy*syz)/dnm
      b01 = (4.0*syz-sy*sz)/dnm
      dz00 = z00 - b00
      dz01 = z01 - (b00+b01*y1)
      dz02 = z02 - (b00+b01*y2)
      dz03 = z03 - (b00+b01*y3)
      volf = dz00**2 + dz01**2 + dz02**2 + dz03**2
      disf = syy

! Calculates the EPSLN value, which is used to decide whether or
! not the volatility factor is essentially zero.
      epsln = (z00**2+z01**2+z02**2+z03**2)*1.0E-12
! Accumulates the weighted primary estimates of zy and their weights.
      IF (volf > epsln) THEN
! - For a finite weight.
        wt = 1.0/ (volf*disf)
        smpef = smpef + wt*pezy
        smwtf = smwtf + wt
      ELSE
! - For an infinite weight.
        smpei = smpei + pezy
        smwti = smwti + 1.0
      END IF
! Saves the necessary values for estimating zxy
      ya(1,ipey) = y1
      ya(2,ipey) = y2
      ya(3,ipey) = y3
      z0ia(1,ipey) = z01
      z0ia(2,ipey) = z02
      z0ia(3,ipey) = z03
      cya(1,ipey) = cy1
      cya(2,ipey) = cy2
      cya(3,ipey) = cy3
      sya(ipey) = sy
      syya(ipey) = syy
      b00ya(ipey) = b00
      b01a(ipey) = b01
    END DO

! Calculates the final estimate of zy.
    IF (smwti < 0.5) THEN
! - When no infinite weights exist.
      zydi = smpef/smwtf
    ELSE
! - When infinite weights exist.
      zydi = smpei/smwti
    END IF
! End of Part 2.

! Part 3.  Estimation of ZXYDI
! Initial setting
    smpef = 0.0
    smwtf = 0.0
    smpei = 0.0
    smwti = 0.0
! Outer DO-loops with respect to the primary estimates in the x direction
    DO  ipex = 1,4
      ix1 = ix0 + idlt(1,ipex)
      ix2 = ix0 + idlt(2,ipex)
      ix3 = ix0 + idlt(3,ipex)
      IF ((ix1 < 1) .OR. (ix2 < 1) .OR. (ix3 < 1) .OR.  &
          (ix1 > nx0) .OR. (ix2 > nx0) .OR. (ix3 > nx0)) CYCLE
! Retrieves the necessary values for estimating zxy in the x direction.
      x1 = xa(1,ipex)
      x2 = xa(2,ipex)
      x3 = xa(3,ipex)
      z10 = zi0a(1,ipex)
      z20 = zi0a(2,ipex)
      z30 = zi0a(3,ipex)
      cx1 = cxa(1,ipex)
      cx2 = cxa(2,ipex)
      cx3 = cxa(3,ipex)
      sx = sxa(ipex)
      sxx = sxxa(ipex)
      b00x = b00xa(ipex)
      b10 = b10a(ipex)

! Inner DO-loops with respect to the primary estimates in the y direction
      DO  ipey = 1,4
        iy1 = iy0 + idlt(1,ipey)
        iy2 = iy0 + idlt(2,ipey)
        iy3 = iy0 + idlt(3,ipey)
        IF ((iy1 < 1) .OR. (iy2 < 1) .OR. (iy3 < 1) .OR. (iy1 > ny0) .OR.  &
            (iy2 > ny0) .OR. (iy3 > ny0)) CYCLE
! Retrieves the necessary values for estimating zxy in the y direction.
        y1 = ya(1,ipey)
        y2 = ya(2,ipey)
        y3 = ya(3,ipey)
        z01 = z0ia(1,ipey)
        z02 = z0ia(2,ipey)
        z03 = z0ia(3,ipey)
        cy1 = cya(1,ipey)
        cy2 = cya(2,ipey)
        cy3 = cya(3,ipey)
        sy = sya(ipey)
        syy = syya(ipey)
        b00y = b00ya(ipey)
        b01 = b01a(ipey)
! Selects and/or supplements the z values.
        IF (nyd >= 4) THEN
          z11 = zd(ix1,iy1)
          z12 = zd(ix1,iy2)
          z13 = zd(ix1,iy3)
          IF (nxd >= 4) THEN
            z21 = zd(ix2,iy1)
            z22 = zd(ix2,iy2)
            z23 = zd(ix2,iy3)
            z31 = zd(ix3,iy1)
            z32 = zd(ix3,iy2)
            z33 = zd(ix3,iy3)
          ELSE IF (nxd == 3) THEN
            z21 = zd(ix2,iy1)
            z22 = zd(ix2,iy2)
            z23 = zd(ix2,iy3)
            z31 = z3f(x1,x2,x3,z01,z11,z21)
            z32 = z3f(x1,x2,x3,z02,z12,z22)
            z33 = z3f(x1,x2,x3,z03,z13,z23)
          ELSE IF (nxd == 2) THEN
            z21 = z2f(x1,x2,z01,z11)
            z22 = z2f(x1,x2,z02,z12)
            z23 = z2f(x1,x2,z03,z13)
            z31 = z2f(x1,x3,z01,z11)
            z32 = z2f(x1,x3,z02,z12)
            z33 = z2f(x1,x3,z03,z13)
          END IF
        ELSE IF (nyd == 3) THEN
          z11 = zd(ix1,iy1)
          z12 = zd(ix1,iy2)
          z13 = z3f(y1,y2,y3,z10,z11,z12)
          IF (nxd >= 4) THEN
            z21 = zd(ix2,iy1)
            z22 = zd(ix2,iy2)
            z31 = zd(ix3,iy1)
            z32 = zd(ix3,iy2)
          ELSE IF (nxd == 3) THEN
            z21 = zd(ix2,iy1)
            z22 = zd(ix2,iy2)
            z31 = z3f(x1,x2,x3,z01,z11,z21)
            z32 = z3f(x1,x2,x3,z02,z12,z22)
          ELSE IF (nxd == 2) THEN
            z21 = z2f(x1,x2,z01,z11)
            z22 = z2f(x1,x2,z02,z12)
            z31 = z2f(x1,x3,z01,z11)
            z32 = z2f(x1,x3,z02,z12)
          END IF
          z23 = z3f(y1,y2,y3,z20,z21,z22)
          z33 = z3f(y1,y2,y3,z30,z31,z32)
        ELSE IF (nyd == 2) THEN
          z11 = zd(ix1,iy1)
          z12 = z2f(y1,y2,z10,z11)
          z13 = z2f(y1,y3,z10,z11)
          IF (nxd >= 4) THEN
            z21 = zd(ix2,iy1)
            z31 = zd(ix3,iy1)
          ELSE IF (nxd == 3) THEN
            z21 = zd(ix2,iy1)
            z31 = z3f(x1,x2,x3,z01,z11,z21)
          ELSE IF (nxd == 2) THEN
            z21 = z2f(x1,x2,z01,z11)
            z31 = z2f(x1,x3,z01,z11)
          END IF
          z22 = z2f(y1,y2,z20,z21)
          z23 = z2f(y1,y3,z20,z21)
          z32 = z2f(y1,y2,z30,z31)
          z33 = z2f(y1,y3,z30,z31)
        END IF
! Calculates the primary estimate of partial derivative zxy as
! the coefficient of the bicubic polynomial.
        dzxy11 = (z11-z10-z01+z00)/ (x1*y1)
        dzxy12 = (z12-z10-z02+z00)/ (x1*y2)
        dzxy13 = (z13-z10-z03+z00)/ (x1*y3)
        dzxy21 = (z21-z20-z01+z00)/ (x2*y1)
        dzxy22 = (z22-z20-z02+z00)/ (x2*y2)
        dzxy23 = (z23-z20-z03+z00)/ (x2*y3)
        dzxy31 = (z31-z30-z01+z00)/ (x3*y1)
        dzxy32 = (z32-z30-z02+z00)/ (x3*y2)
        dzxy33 = (z33-z30-z03+z00)/ (x3*y3)
        pezxy = cx1* (cy1*dzxy11+cy2*dzxy12+cy3*dzxy13) +  &
                cx2* (cy1*dzxy21+cy2*dzxy22+cy3*dzxy23) +  &
                cx3* (cy1*dzxy31+cy2*dzxy32+cy3*dzxy33)
! Calculates the volatility factor and distance factor in the x
! and y directions for the primary estimate of zxy.
        b00 = (b00x+b00y)/2.0
        sxy = sx*sy
        sxxy = sxx*sy
        sxyy = sx*syy
        sxxyy = sxx*syy
        sxyz = x1* (y1*z11+y2*z12+y3*z13) + x2* (y1*z21+y2*z22+y3*z23) +  &
               x3* (y1*z31+y2*z32+y3*z33)
        b11 = (sxyz-b00*sxy-b10*sxxy-b01*sxyy)/sxxyy
        dz00 = z00 - b00
        dz01 = z01 - (b00+b01*y1)
        dz02 = z02 - (b00+b01*y2)
        dz03 = z03 - (b00+b01*y3)
        dz10 = z10 - (b00+b10*x1)
        dz11 = z11 - (b00+b01*y1+x1* (b10+b11*y1))
        dz12 = z12 - (b00+b01*y2+x1* (b10+b11*y2))
        dz13 = z13 - (b00+b01*y3+x1* (b10+b11*y3))
        dz20 = z20 - (b00+b10*x2)
        dz21 = z21 - (b00+b01*y1+x2* (b10+b11*y1))
        dz22 = z22 - (b00+b01*y2+x2* (b10+b11*y2))
        dz23 = z23 - (b00+b01*y3+x2* (b10+b11*y3))
        dz30 = z30 - (b00+b10*x3)
        dz31 = z31 - (b00+b01*y1+x3* (b10+b11*y1))
        dz32 = z32 - (b00+b01*y2+x3* (b10+b11*y2))
        dz33 = z33 - (b00+b01*y3+x3* (b10+b11*y3))
        volf = dz00**2 + dz01**2 + dz02**2 + dz03**2 +  &
            dz10**2 + dz11**2 + dz12**2 + dz13**2 +  &
            dz20**2 + dz21**2 + dz22**2 + dz23**2 +  &
            dz30**2 + dz31**2 + dz32**2 + dz33**2
        disf = sxx*syy
! Calculates EPSLN.
        epsln = (z00**2 + z01**2 + z02**2 + z03**2 + z10**2 +   &
                z11**2 + z12**2 + z13**2 + z20**2 + z21**2 + z22**2 +   &
                z23**2 + z30**2 + z31**2 + z32**2 + z33**2)* 1.0E-12
! Accumulates the weighted primary estimates of zxy and their weights.
        IF (volf > epsln) THEN
! - For a finite weight.
          wt = 1.0/ (volf*disf)
          smpef = smpef + wt*pezxy
          smwtf = smwtf + wt
        ELSE
! - For an infinite weight.
          smpei = smpei + pezxy
          smwti = smwti + 1.0
        END IF
      END DO
    END DO

! Calculates the final estimate of zxy.
    IF (smwti < 0.5) THEN
! - When no infinite weights exist.
      zxydi = smpef/smwtf
    ELSE
! - When infinite weights exist.
      zxydi = smpei/smwti
    END IF
! End of Part 3

    pdd(1,ix0,iy0) = zxdi
    pdd(2,ix0,iy0) = zydi
    pdd(3,ix0,iy0) = zxydi
  END DO
END DO
RETURN
END SUBROUTINE rgpd3p



SUBROUTINE rglctn(nxd, nyd, xd, yd, nip, xi, yi, inxi, inyi)

! Location of the desired points in a rectangular grid
! (a supporting subroutine of the RGBI3P/RGSF3P subroutine package)

! Hiroshi Akima
! U.S. Department of Commerce, NTIA/ITS
! Version of 1995/08

! This subroutine locates the desired points in a rectangular grid
! in the x-y plane.

! The grid lines can be unevenly spaced.

! The input arguments are
!   NXD  = number of the input-grid data points in the x
!          coordinate (must be 2 or greater),
!   NYD  = number of the input-grid data points in the y
!          coordinate (must be 2 or greater),
!   XD   = array of dimension NXD containing the x coordinates of the
!          input-grid data points (must be in a monotonic increasing order),
!   YD   = array of dimension NYD containing the y coordinates of the
!          input-grid data points (must be in a monotonic increasing order),
!   NIP  = number of the output points to be located (must be 1 or greater),
!   XI   = array of dimension NIP containing the x coordinates
!          of the output points to be located,
!   YI   = array of dimension NIP containing the y coordinates
!          of the output points to be located.

! The output arguments are
!   INXI = integer array of dimension NIP where the interval
!          numbers of the XI array elements are to be stored,
!   INYI = integer array of dimension NIP where the interval
!          numbers of the YI array elements are to be stored.
! The interval numbers are between 0 and NXD and between 0 and NYD,
! respectively.


! Specification statements
!     .. Scalar Arguments ..

INTEGER, INTENT(IN)   :: nxd
INTEGER, INTENT(IN)   :: nyd
DOUBLE PRECISION, INTENT(IN)      :: xd(nxd)
DOUBLE PRECISION, INTENT(IN)      :: yd(nyd)
INTEGER, INTENT(IN)   :: nip
DOUBLE PRECISION, INTENT(IN)      :: xi(nip)
DOUBLE PRECISION, INTENT(IN)      :: yi(nip)
INTEGER, INTENT(OUT)  :: inxi(nip)
INTEGER, INTENT(OUT)  :: inyi(nip)

!     ..
!     .. Local Scalars ..
DOUBLE PRECISION     :: xii, yii
INTEGER  :: iip, imd, imn, imx, ixd, iyd, nintx, ninty

!     ..
! DO-loop with respect to IIP, which is the point number of the output point
DO  iip = 1,nip
  xii = xi(iip)
  yii = yi(iip)
! Checks if the x coordinate of the IIPth output point, XII, is
! in a new interval.  (NINTX is the new-interval flag.)
  IF (iip == 1) THEN
    nintx = 1
  ELSE
    nintx = 0
    IF (ixd == 0) THEN
      IF (xii > xd(1)) nintx = 1
    ELSE IF (ixd < nxd) THEN
      IF ((xii < xd(ixd)) .OR. (xii > xd(ixd+1))) nintx = 1
    ELSE
      IF (xii < xd(nxd)) nintx = 1
    END IF
  END IF

! Locates the output point by binary search if XII is in a new interval.
! Determines IXD for which XII lies between XD(IXD) and XD(IXD+1).
  IF (nintx == 1) THEN
    IF (xii <= xd(1)) THEN
      ixd = 0
    ELSE IF (xii < xd(nxd)) THEN
      imn = 1
      imx = nxd
      imd = (imn+imx)/2
      10 IF (xii >= xd(imd)) THEN
        imn = imd
      ELSE
        imx = imd
      END IF
      imd = (imn+imx)/2
      IF (imd > imn) GO TO 10
      ixd = imd
    ELSE
      ixd = nxd
    END IF
  END IF
  inxi(iip) = ixd

! Checks if the y coordinate of the IIPth output point, YII, is
! in a new interval.  (NINTY is the new-interval flag.)
  IF (iip == 1) THEN
    ninty = 1
  ELSE
    ninty = 0
    IF (iyd == 0) THEN
      IF (yii > yd(1)) ninty = 1
    ELSE IF (iyd < nyd) THEN
      IF ((yii < yd(iyd)) .OR. (yii > yd(iyd+1))) ninty = 1
    ELSE
      IF (yii < yd(nyd)) ninty = 1
    END IF
  END IF

! Locates the output point by binary search if YII is in a new interval.
! Determines IYD for which YII lies between YD(IYD) and YD(IYD+1).
  IF (ninty == 1) THEN
    IF (yii <= yd(1)) THEN
      iyd = 0
    ELSE IF (yii < yd(nyd)) THEN
      imn = 1
      imx = nyd
      imd = (imn+imx)/2
      20 IF (yii >= yd(imd)) THEN
        imn = imd
      ELSE
        imx = imd
      END IF
      imd = (imn+imx)/2
      IF (imd > imn) GO TO 20
      iyd = imd
    ELSE
      iyd = nyd
    END IF
  END IF
  inyi(iip) = iyd
END DO
RETURN
END SUBROUTINE rglctn



SUBROUTINE rgplnl(nxd, nyd, xd, yd, zd, pdd, nip, xi, yi, inxi, inyi, zi)

! Polynomials for rectangular-grid bivariate interpolation and surface fitting
! (a supporting subroutine of the RGBI3P/RGSF3P subroutine package)

! Hiroshi Akima
! U.S. Department of Commerce, NTIA/ITS
! Version of 1995/08

! This subroutine determines a polynomial in x and y for a rectangle of the
! input grid in the x-y plane and calculates the z value for the desired
! points by evaluating the polynomial for rectangular-grid bivariate
! interpolation and surface fitting.

! The input arguments are
!   NXD  = number of the input-grid data points in the x
!          coordinate (must be 2 or greater),
!   NYD  = number of the input-grid data points in the y
!          coordinate (must be 2 or greater),
!   XD   = array of dimension NXD containing the x coordinates of the
!          input-grid data points (must be in a monotonic increasing order),
!   YD   = array of dimension NYD containing the y coordinates of the
!          input-grid data points (must be in a monotonic increasing order),
!   ZD   = two-dimensional array of dimension NXD*NYD
!          containing the z(x,y) values at the input-grid data points,
!   PDD  = three-dimensional array of dimension 3*NXD*NYD
!          containing the estimated zx, zy, and zxy values
!          at the input-grid data points,
!   NIP  = number of the output points at which interpolation
!          is to be performed,
!   XI   = array of dimension NIP containing the x coordinates
!          of the output points,
!   YI   = array of dimension NIP containing the y coordinates
!          of the output points,
!   INXI = integer array of dimension NIP containing the
!          interval numbers of the input grid intervals in the
!          x direction where the x coordinates of the output points lie,
!   INYI = integer array of dimension NIP containing the
!          interval numbers of the input grid intervals in the
!          y direction where the y coordinates of the output points lie.

! The output argument is
!   ZI   = array of dimension NIP, where the interpolated z
!          values at the output points are to be stored.


! Specification statements
!     .. Scalar Arguments ..

INTEGER, INTENT(IN)  :: nxd
INTEGER, INTENT(IN)  :: nyd
DOUBLE PRECISION, INTENT(IN)     :: xd(nxd)
DOUBLE PRECISION, INTENT(IN)     :: yd(nyd)
DOUBLE PRECISION, INTENT(IN)     :: zd(nxd,nyd)
DOUBLE PRECISION, INTENT(IN)     :: pdd(3,nxd,nyd)
INTEGER, INTENT(IN)  :: nip
DOUBLE PRECISION, INTENT(IN)     :: xi(nip)
DOUBLE PRECISION, INTENT(IN)     :: yi(nip)
INTEGER, INTENT(IN)  :: inxi(nip)
INTEGER, INTENT(IN)  :: inyi(nip)
DOUBLE PRECISION, INTENT(OUT)    :: zi(nip)

!     ..
!     .. Local Scalars ..
DOUBLE PRECISION :: a, b, c, d, dx, dxsq, dy, dysq, p00, p01, p02, p03, p10, p11,  &
        p12, p13, p20, p21, p22, p23, p30, p31, p32, p33, q0, q1, q2,  &
        q3, u, v, x0, xii, y0, yii, z00, z01, z0dx, z0dy, z10, z11,  &
        z1dx, z1dy, zdxdy, zii, zx00, zx01, zx0dy, zx10, zx11,  &
        zx1dy, zxy00, zxy01, zxy10, zxy11, zy00, zy01, zy0dx, zy10, zy11, zy1dx
INTEGER :: iip, ixd0, ixd1, ixdi, ixdipv, iyd0, iyd1, iydi, iydipv
!     ..
!     .. Intrinsic Functions ..
! INTRINSIC        MAX
!     ..

! Calculation
! Outermost DO-loop with respect to the output point
DO  iip = 1,nip
  xii = xi(iip)
  yii = yi(iip)
  IF (iip == 1) THEN
    ixdipv = -1
    iydipv = -1
  ELSE
    ixdipv = ixdi
    iydipv = iydi
  END IF
  ixdi = inxi(iip)
  iydi = inyi(iip)

! Retrieves the z and partial derivative values at the origin of
! the coordinate for the rectangle.
  IF (ixdi /= ixdipv .OR. iydi /= iydipv) THEN
    ixd0 = MAX(1,ixdi)
    iyd0 = MAX(1,iydi)
    x0 = xd(ixd0)
    y0 = yd(iyd0)
    z00 = zd(ixd0,iyd0)
    zx00 = pdd(1,ixd0,iyd0)
    zy00 = pdd(2,ixd0,iyd0)
    zxy00 = pdd(3,ixd0,iyd0)
  END IF

! Case 1.  When the rectangle is inside the data area in both the
! x and y directions.
  IF ((ixdi > 0 .AND. ixdi < nxd) .AND. (iydi > 0 .AND. iydi < nyd)) THEN
! Retrieves the z and partial derivative values at the other three
! vertices of the rectangle.
    IF (ixdi /= ixdipv .OR. iydi /= iydipv) THEN
      ixd1 = ixd0 + 1
      dx = xd(ixd1) - x0
      dxsq = dx*dx
      iyd1 = iyd0 + 1
      dy = yd(iyd1) - y0
      dysq = dy*dy
      z10 = zd(ixd1,iyd0)
      z01 = zd(ixd0,iyd1)
      z11 = zd(ixd1,iyd1)
      zx10 = pdd(1,ixd1,iyd0)
      zx01 = pdd(1,ixd0,iyd1)
      zx11 = pdd(1,ixd1,iyd1)
      zy10 = pdd(2,ixd1,iyd0)
      zy01 = pdd(2,ixd0,iyd1)
      zy11 = pdd(2,ixd1,iyd1)
      zxy10 = pdd(3,ixd1,iyd0)
      zxy01 = pdd(3,ixd0,iyd1)
      zxy11 = pdd(3,ixd1,iyd1)
! Calculates the polynomial coefficients.
      z0dx = (z10-z00)/dx
      z1dx = (z11-z01)/dx
      z0dy = (z01-z00)/dy
      z1dy = (z11-z10)/dy
      zx0dy = (zx01-zx00)/dy
      zx1dy = (zx11-zx10)/dy
      zy0dx = (zy10-zy00)/dx
      zy1dx = (zy11-zy01)/dx
      zdxdy = (z1dy-z0dy)/dx
      a = zdxdy - zx0dy - zy0dx + zxy00
      b = zx1dy - zx0dy - zxy10 + zxy00
      c = zy1dx - zy0dx - zxy01 + zxy00
      d = zxy11 - zxy10 - zxy01 + zxy00
      p00 = z00
      p01 = zy00
      p02 = (2.0* (z0dy-zy00)+z0dy-zy01)/dy
      p03 = (-2.0*z0dy+zy01+zy00)/dysq
      p10 = zx00
      p11 = zxy00
      p12 = (2.0* (zx0dy-zxy00)+zx0dy-zxy01)/dy
      p13 = (-2.0*zx0dy+zxy01+zxy00)/dysq
      p20 = (2.0* (z0dx-zx00)+z0dx-zx10)/dx
      p21 = (2.0* (zy0dx-zxy00)+zy0dx-zxy10)/dx
      p22 = (3.0* (3.0*a-b-c)+d)/ (dx*dy)
      p23 = (-6.0*a+2.0*b+3.0*c-d)/ (dx*dysq)
      p30 = (-2.0*z0dx+zx10+zx00)/dxsq
      p31 = (-2.0*zy0dx+zxy10+zxy00)/dxsq
      p32 = (-6.0*a+3.0*b+2.0*c-d)/ (dxsq*dy)
      p33 = (2.0* (2.0*a-b-c)+d)/ (dxsq*dysq)
    END IF

! Evaluates the polynomial.
    u = xii - x0
    v = yii - y0
    q0 = p00 + v* (p01+v* (p02+v*p03))
    q1 = p10 + v* (p11+v* (p12+v*p13))
    q2 = p20 + v* (p21+v* (p22+v*p23))
    q3 = p30 + v* (p31+v* (p32+v*p33))
    zii = q0 + u* (q1+u* (q2+u*q3))
! End of Case 1

! Case 2.  When the rectangle is inside the data area in the x
! direction but outside in the y direction.
  ELSE IF ((ixdi > 0.AND.ixdi < nxd) .AND.  &
        (iydi <= 0.OR.iydi >= nyd)) THEN
! Retrieves the z and partial derivative values at the other
! vertex of the semi-infinite rectangle.
    IF (ixdi /= ixdipv .OR. iydi /= iydipv) THEN
      ixd1 = ixd0 + 1
      dx = xd(ixd1) - x0
      dxsq = dx*dx
      z10 = zd(ixd1,iyd0)
      zx10 = pdd(1,ixd1,iyd0)
      zy10 = pdd(2,ixd1,iyd0)
      zxy10 = pdd(3,ixd1,iyd0)
! Calculates the polynomial coefficients.
      z0dx = (z10-z00)/dx
      zy0dx = (zy10-zy00)/dx
      p00 = z00
      p01 = zy00
      p10 = zx00
      p11 = zxy00
      p20 = (2.0* (z0dx-zx00)+z0dx-zx10)/dx
      p21 = (2.0* (zy0dx-zxy00)+zy0dx-zxy10)/dx
      p30 = (-2.0*z0dx+zx10+zx00)/dxsq
      p31 = (-2.0*zy0dx+zxy10+zxy00)/dxsq
    END IF
! Evaluates the polynomial.
    u = xii - x0
    v = yii - y0
    q0 = p00 + v*p01
    q1 = p10 + v*p11
    q2 = p20 + v*p21
    q3 = p30 + v*p31
    zii = q0 + u* (q1+u* (q2+u*q3))
! End of Case 2

! Case 3.  When the rectangle is outside the data area in the x
! direction but inside in the y direction.
  ELSE IF ((ixdi <= 0.OR.ixdi >= nxd) .AND.  &
        (iydi > 0 .AND. iydi < nyd)) THEN
! Retrieves the z and partial derivative values at the other
! vertex of the semi-infinite rectangle.
    IF (ixdi /= ixdipv .OR. iydi /= iydipv) THEN
      iyd1 = iyd0 + 1
      dy = yd(iyd1) - y0
      dysq = dy*dy
      z01 = zd(ixd0,iyd1)
      zx01 = pdd(1,ixd0,iyd1)
      zy01 = pdd(2,ixd0,iyd1)
      zxy01 = pdd(3,ixd0,iyd1)
! Calculates the polynomial coefficients.
      z0dy = (z01-z00)/dy
      zx0dy = (zx01-zx00)/dy
      p00 = z00
      p01 = zy00
      p02 = (2.0*(z0dy-zy00)+z0dy-zy01)/dy
      p03 = (-2.0*z0dy+zy01+zy00)/dysq
      p10 = zx00
      p11 = zxy00
      p12 = (2.0*(zx0dy-zxy00) + zx0dy - zxy01)/dy
      p13 = (-2.0*zx0dy + zxy01 + zxy00)/dysq
    END IF

! Evaluates the polynomial.
    u = xii - x0
    v = yii - y0
    q0 = p00 + v* (p01 + v*(p02+v*p03))
    q1 = p10 + v* (p11 + v*(p12+v*p13))
    zii = q0 + u*q1
! End of Case 3

! Case 4.  When the rectangle is outside the data area in both the
! x and y direction.
  ELSE IF ((ixdi <= 0 .OR. ixdi >= nxd) .AND.  &
           (iydi <= 0 .OR. iydi >= nyd)) THEN
! Calculates the polynomial coefficients.
    IF (ixdi /= ixdipv .OR. iydi /= iydipv) THEN
      p00 = z00
      p01 = zy00
      p10 = zx00
      p11 = zxy00
    END IF
! Evaluates the polynomial.
    u = xii - x0
    v = yii - y0
    q0 = p00 + v*p01
    q1 = p10 + v*p11
    zii = q0 + u*q1
  END IF
! End of Case 4
  zi(iip) = zii
END DO

RETURN
END SUBROUTINE rgplnl

END MODULE Grid_Interpolation